If, #|x+5|<1/2,# then show that#|x^2-25|<21/4# Could someone help me to solve the problem?

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Answer 1 · 2018-03-06T17:46:30

Kindly refer to the Explanation.

#"Prerequisites : "(1) : |a+b| le |a|+|b|, (2) : |ab|=|a||b|#.

#"By (2), "|x^2-25|=|(x+5)(x-5)|=|x+5||x-5|#.

#"Now, "|x+5| lt 1/2............[Given]#.

Multiplying by #|x-5| gt 0#, we get,

# |x+5||x-5| lt 1/2|x-5|............(ast1)#.

But, #|x-5|=|(x+5)+(-10)|#,

# le |x+5|+|(-10)|...[because, (1)]#,

#=|x+5|+10 lt 1/2+10........[because," Given]"#.

# rArr |x-5| le 21/2............(ast2)#.

#(ast1) and (ast2) rArr |x+5||x-5| lt 1/2*21/2, or, #

# |x^2-25| lt 21/4#, as desired!